ARTICLE IN PRESS International Journal of Non-Linear Mechanics ] (]]]]) ]]]–]]] Contents lists available at ScienceDirect International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm Discontinuous synchrony in an array of Van der Pol oscillators Przemyslaw Perlikowski a,b, Andrzej Stefanski a,Ã, Tomasz Kapitaniak a a Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland b Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany article info abstract Article history: We show the phenomenon of complete synchronization in an network of coupled oscillators. We Received 24 November 2008 confirm that non-diagonal coupling can lead to the appearance or disappearance of synchronous Received in revised form windows (ragged synchronizability phenomenon) in the coupling parameter space. We also show the 27 November 2009 appearance of clusters (synchronization in one or more group) between coupled systems. Our Accepted 28 January 2010 numerical studies are confirmed by an electronic experiment. & 2010 Elsevier Ltd. All rights reserved. Keywords: Complete synchronization Van der Pol oscillator Ragged synchronizability Clusters 1. Introduction meter space, when the number of oscillators in the array or topology of connections between them changes. This phenomen- Collective dynamics of coupled systems has been known for a on has been called the ragged synchronizability (RSA). The long time, i.e., since the second half of 17th century, when existence of the RSA has been confirmed numerically [23] and Huygens discovered the synchronization of two clock pendulums experimentally [17]. Usually, in papers concerning these new [10]. Next, this phenomenon has been observed and investigated ideas classical autonomous dynamical systems have been used as in various types of mechanical or electrical systems [5,19,24]. the examples, i.e., the Lorenz and Rossler¨ oscillators, non-linear Recently, the idea of synchronization has been also adopted for discrete-time systems (e.g. logistic and Henon map), neuron chaotic systems [15] and it has become an object of great interest models etc. On the other hand there is a lack of works on new in many areas of science, e.g., biology [9,22], communication [6] synchronization concepts applied for non-autonomous systems or laser physics [28,32]. like mechanical or electrical oscillators with external forcing. Through the last years a number of new types of synchroniza- Hence, this paper deals with synchronization of non-linear tion have been identified [20,21] and new interesting ideas have oscillators with a harmonic driving (Section 5), where phenomena appeared [3,27]. It has been demonstrated that two or more of the RSA and clustering (synchronization of oscillator’s sub- chaotic systems can synchronize if they are under control of some groups) can be observed. For needs of clear presentation of the connection mechanisms. Typical examples of such mechanisms RSA effect we concentrated on the complete synchronization (CS) are drive-response [15,16] or active-passive decomposition [13] mode in sense of total time-coincidence of phases and ampli- of the considered system, direct bi-directional (mutual) or tudes. The synchronization of periodic responses with phase shift unidirectional (master-slave) diffusive coupling between the is represented here by effect of clustering. In the numerical oscillators [12,29,30]. analysis, the Van der Pol (VdP) [26] oscillator with non-linear In our previous work [23], we presented an example of non- damping, which can be treated as an equivalent of mechanical diagonally coupled array of Duffing oscillators, in which multiple self-excited system, has been applied as an array node system. disconnected synchronous regions of coupling strength occur. The Such a choice is due to option of continuous control of the system term non-diagonal coupling means that the network nodes are parameters in electrical oscillators. Then the experimental linked with others via non-diagonal components of linking bifurcational analysis of the system dynamics is possible. (output) function. We have also observed the appearance or This paper is organized as follows. In Section 2 basic disappearance of such synchronous windows in coupling para- definitions concerning synchronization problems under consid- eration are presented. Section 3 approaches the idea of Master Stability Function (MSF) [14], which is a main tool for carried out à Corresponding author. analysis of the synchronization. Next Section 4 contains an E-mail address: [email protected] (A. Stefanski). detailed description of investigated VdP oscillator. Results of 0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2010.01.010 Please cite this article as: P. Perlikowski, et al., Discontinuous synchrony in an array of Van der Pol oscillators, Int. J. Non-Linear Mech. (2010), doi:10.1016/j.ijnonlinmec.2010.01.010 ARTICLE IN PRESS 2 P. Perlikowski et al. / International Journal of Non-Linear Mechanics ] (]]]]) ]]]–]]] numerical and experimental study and their comparison are After the derivation the variational equation of the network demonstrated in Section 5. The paper finishes with conclusions system (4) is as follows (Section 6). n_ ¼½I DFþsðG DHÞx, ð5Þ 2. Synchronization and clustering where n ¼ðx1,x2, ...,xNÞ represents collection of perturbation, I is an identity matrix and DF, DH are Jacobi matrices of system (4) and output function, respectively. Pecora and Carroll [15] defined the CS between two dynamical After the block diagonalization of the variational equation systems as a state when their state trajectories x(t) and y(t) (Eq. (5)) we have converge to the same values and continue in such relation further in time. Thus, the CS means a full coincidence of phases _ xk ¼½Df þsg DHxk, ð6Þ (frequencies) and amplitudes of the systems response. Earlier k the problem of the CS threshold in arrays of coupled identical For g0 ¼ 0 we have linearized the equation of the node system oscillators was studied by Fujisaka and Yamada [8,30] and (Eq. (3)) which is corresponding to the mode longitudinal to Pikovsky [18]. invariant synchronization manifold x1 ¼ x2 ¼ÁÁ¼xN. The re- maining NÀ1 eigenvalues represent different transverse modes of Definition 1.1. Complete synchronization of two dynamical perturbation from synchronous state. In the general case they can systems represented with their phase plane trajectories x(t) and be complex numbers. However, for identical coupled mechanical y(t), respectively, takes place when the following relation is oscillators eigenvalues gk are real numbers due to the symmetry fulfilled: of coupling. This symmetry results from a mutual character of the limJxðtÞyðtÞJ ¼ 0: ð1Þ interaction between mechanical systems according to 3rd - t 1 Newton’s law of dynamics. Such a symmetry of coupling (i.e. diagonal symmetry of connectivity matrix G) results from It is also described in the subject literature as identical or full the identity of coupled oscillators. In case of non-identical synchronization [16,20]. The CS state can be reached only when systems (e.g. different masses of the oscillators) there appear two identical dynamical systems are concerned, say, they are non-symmetry in matrix G, in spite of mutual interaction of given with the same ODEs with identical system parameters. This oscillators. condition of identity may not be fulfilled due to presence of an Assuming that g represents an arbitrary value of g and f external noise or parameters mismatch what usually can happen k symbolizes an arbitrary transverse mode x , we can define the in real systems. If scale of such disturbances is relatively small, k generic variational equation for any node system then both systems may eventually reach a state called imperfect complete synchronization (ICS) [12], sometimes named as z_ ¼½Df þsgDHz: ð7Þ practical or disturbed synchronization. Generic variational equation (Eq. (7)) describes an evolution of Definition 1.2. Imperfect complete synchronization of two any perturbation in the directions transversal to the final dynamical systems represented with their phase plane trajec- synchronous state, that dynamics is governed by Eq. (3). Now, tories x(t) and y(t), respectively, occurs when the following we can obtain the MSF for the considered case as the largest inequality is fulfilled 1 transversal Lyapunov exponent lT , calculated for generic varia- limJxðtÞyðtÞJ e, ð2Þ tional equation (Eq. (7)), including the solution of Eq. (3), in - r t 1 function of the product sg. An exemplary MSF graph is depicted in where e is a small parameter. The CS (or the ICS) of entire network Figs. 1a and b. If discrete spectrum of products sgk corresponding or array of mechanical oscillators means a collective motion of to all transversal eigenmodes can be found in the ranges of them, i.e., they have the same position and velocity in each negative transversal Lyapunov exponent (see Fig. 1a) then moment of time evolution. However, if there is a set composed of synchronous state is stable for the analyzed configuration of N 42 identical nodes then it can be divided into two or more couplings. This is a necessary condition for synchronization of all subsets within which the motion of oscillators is collective while network nodes. However, sometimes it may not be a sufficient between subsets the dynamics is uncorrelated or at least a phase one due to possible local instability of the synchronization shift is observed, e.g. two groups of subsystems in anti-phase manifold [2]. On the other hand, if even one of values sgk is 1 regime. Such subgroups of synchronized oscillators are called located in the range of positive lT (see Fig. 1b) then the CS clusters [4,11]. of all oscillators is impossible, but synchronization in clusters can occur. 3. Stability of synchronous state Consider a dynamical system 4. Analyzed oscillator x_ ¼ fðxÞ, ð3Þ where xARm. A graphical representation of the analyzed mechanical oscil- The dynamics of any network of N identical oscillators can be lator is shown in Fig.